Complex bandpass (BP) active filters are widely used in integrated receivers. These filters serve primarily as intermediate-frequency (IF) channel select filters with an additional function of providing image rejection. Their ability to reject unwanted image frequencies of the preceding mixer results directly from their non-symmetrical transfer characteristic. Depending on its input signal conditioning, a complex BP filter transmits for positive frequencies and rejects all negative frequencies, or vice-versa, the filter transmits for negative frequencies and blocks all positive frequencies.
The achievable image-rejection ratio (IRR) depends on the matching of on-chip components used in the complex filter. These components include resistors, capacitors and transconductors. Also, non-ideal gain of operational amplifiers (opamps) if used for filter synthesis results in IRR degradation. The IRR performance also depends on the choice of synthesis method. Certain methods are more sensitive to component variation than others. Practically achievable IRR of a complex filter is 30–35 dB. If extreme caution is taken to achieve an excellent component matching, or if an IRR is enhanced by a special automatic tuning scheme IRR better than 55 dB is achievable.
Complex BP filters can be realized using two distinct synthesis methods: the active ladder simulation and the direct synthesis. Similarly to the classical passive LC ladder synthesis method, in the active ladder simulation the pole frequency and its quality (Q) factor are defined by all filter elements. Contrary, in the direct synthesis method the pole frequency and its Q factor are defined by elements of one particular filter section. Due to its lower sensitivity to the component value variation, the active ladder simulation method is superior to the direct synthesis method. However, the latter results in a simple circuit that is usually easier to integrate.
The complex bandpass filters can also be categorized according to the chosen active synthesis method. Two different active synthesis techniques have been used: the active-RC technique described in U.S. Pat. No. 4,914,408 and the gyrator method described in U.S. Pat. No. 6,346,850. In the active-RC method the transfer function is realized using active-RC integrators built with input series resistors and feedback capacitors around opamps. The gyrator method uses voltage-controlled current sources and capacitors to realize integrators. The advantage of the gyrator method over the active-RC method stems from the gyrator method's ability to adjust the filter pole frequency through adjusting the transconductance of voltage controlled current sources, which is not easily achievable in a R and C arrangement of the active RC filters.
Due to their prime application as channel select filters complex BP filters must demonstrate sharp roll-off outside their pass-bands. In wireless receiver system design, BP filter attenuation determines such critical parameters as co-channel and adjacent channel rejection. Steep roll-off is not easily achievable with all-pole transfer functions. Depending on their order all-pole transfer functions may be quite steep, but as illustrated in FIG. 7, their roll-off is never as steep as that of filters that contain transmission zeros in their transfer function. For all these reasons, in an integrated receiver design there is a strong need for complex BP filters with transmission zeros.